According to the tabloid, Carroll has attracted interest from a range of Premier League clubs.
However, West Ham are determined to keep hold of the England international, as they feel he could hit a large amount of goals next term.
“Andy has just had the best six months he has ever had for the club and now he has had a couple of operations he should be over all his injuries,” Sullivan said.
“So hopefully he should have an injury-free season and get 15-20 goals.”
Carroll moved to West Ham on a permanent transfer back in 2013, and his time at the London club has been disrupted due to injury so far.
Carroll did manage to play 27 times last season in the Premier League, however, and he hit the back of the net on nine occasions.
Slaven Bilic has generally used Carroll as a rotational option during his time in charge, with the former Newcastle United man often used as an impact substitute.
Carroll could face more competition for a place in West Ham’s team next season, as Bilic is preparing to invest in a striker.
The Hammers are desperate to add another forward to their group, but their attempts have proved unsuccessful so far.